Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑥 = ∅ → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ ∅ ) ) |
2 |
1
|
eleq2d |
⊢ ( 𝑥 = ∅ → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 𝐹 ∈ ( Fmla ‘ ∅ ) ) ) |
3 |
|
fveq2 |
⊢ ( 𝑥 = ∅ → ( ( ∅ Sat ∅ ) ‘ 𝑥 ) = ( ( ∅ Sat ∅ ) ‘ ∅ ) ) |
4 |
3
|
eleq2d |
⊢ ( 𝑥 = ∅ → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ ∅ ) ) ) |
5 |
2 4
|
bibi12d |
⊢ ( 𝑥 = ∅ → ( ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ↔ ( 𝐹 ∈ ( Fmla ‘ ∅ ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ ∅ ) ) ) ) |
6 |
5
|
imbi2d |
⊢ ( 𝑥 = ∅ → ( ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ) ↔ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ ∅ ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ ∅ ) ) ) ) ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ 𝑦 ) ) |
8 |
7
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 𝐹 ∈ ( Fmla ‘ 𝑦 ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( ∅ Sat ∅ ) ‘ 𝑥 ) = ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) |
10 |
9
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) |
11 |
8 10
|
bibi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ↔ ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ) ↔ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ) |
13 |
|
fveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ suc 𝑦 ) ) |
14 |
13
|
eleq2d |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ) ) |
15 |
|
fveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( ( ∅ Sat ∅ ) ‘ 𝑥 ) = ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ) |
16 |
15
|
eleq2d |
⊢ ( 𝑥 = suc 𝑦 → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ) ) |
17 |
14 16
|
bibi12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ↔ ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ) ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ) ↔ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ) ) ) ) |
19 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ 𝑁 ) ) |
20 |
19
|
eleq2d |
⊢ ( 𝑥 = 𝑁 → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 𝐹 ∈ ( Fmla ‘ 𝑁 ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( ( ∅ Sat ∅ ) ‘ 𝑥 ) = ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) |
22 |
21
|
eleq2d |
⊢ ( 𝑥 = 𝑁 → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) |
23 |
20 22
|
bibi12d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ↔ ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) ) |
24 |
23
|
imbi2d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑥 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑥 ) ) ) ↔ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) ) ) |
25 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐹 → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
26 |
25
|
2rexbidv |
⊢ ( 𝑥 = 𝐹 → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
27 |
26
|
elrab |
⊢ ( 𝐹 ∈ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } ↔ ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
28 |
|
eqidd |
⊢ ( ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) → ∅ = ∅ ) |
29 |
|
simpr |
⊢ ( ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) |
30 |
28 29
|
jca |
⊢ ( ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) → ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
31 |
|
simpr |
⊢ ( ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) |
32 |
31
|
anim2i |
⊢ ( ( 𝐹 ∈ V ∧ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
33 |
32
|
ex |
⊢ ( 𝐹 ∈ V → ( ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) → ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
34 |
30 33
|
impbid2 |
⊢ ( 𝐹 ∈ V → ( ( 𝐹 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
35 |
27 34
|
syl5bb |
⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
36 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } |
37 |
36
|
eleq2i |
⊢ ( 𝐹 ∈ ( Fmla ‘ ∅ ) ↔ 𝐹 ∈ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } ) |
38 |
37
|
a1i |
⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ ∅ ) ↔ 𝐹 ∈ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } ) ) |
39 |
|
satf00 |
⊢ ( ( ∅ Sat ∅ ) ‘ ∅ ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } |
40 |
39
|
a1i |
⊢ ( 𝐹 ∈ V → ( ( ∅ Sat ∅ ) ‘ ∅ ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } ) |
41 |
40
|
eleq2d |
⊢ ( 𝐹 ∈ V → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ ∅ ) ↔ 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } ) ) |
42 |
|
0ex |
⊢ ∅ ∈ V |
43 |
|
eqeq1 |
⊢ ( 𝑦 = ∅ → ( 𝑦 = ∅ ↔ ∅ = ∅ ) ) |
44 |
43 26
|
bi2anan9r |
⊢ ( ( 𝑥 = 𝐹 ∧ 𝑦 = ∅ ) → ( ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
45 |
44
|
opelopabga |
⊢ ( ( 𝐹 ∈ V ∧ ∅ ∈ V ) → ( 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
46 |
42 45
|
mpan2 |
⊢ ( 𝐹 ∈ V → ( 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
47 |
41 46
|
bitrd |
⊢ ( 𝐹 ∈ V → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ ∅ ) ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝐹 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
48 |
35 38 47
|
3bitr4d |
⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ ∅ ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ ∅ ) ) ) |
49 |
|
eqid |
⊢ ∅ = ∅ |
50 |
49
|
biantrur |
⊢ ( ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
51 |
50
|
bicomi |
⊢ ( ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ↔ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
52 |
51
|
a1i |
⊢ ( 𝐹 ∈ V → ( ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ↔ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
53 |
|
eqeq1 |
⊢ ( 𝑧 = ∅ → ( 𝑧 = ∅ ↔ ∅ = ∅ ) ) |
54 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐹 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
55 |
54
|
rexbidv |
⊢ ( 𝑥 = 𝐹 → ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
56 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐹 → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
57 |
56
|
rexbidv |
⊢ ( 𝑥 = 𝐹 → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
58 |
55 57
|
orbi12d |
⊢ ( 𝑥 = 𝐹 → ( ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
59 |
58
|
rexbidv |
⊢ ( 𝑥 = 𝐹 → ( ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
60 |
53 59
|
bi2anan9r |
⊢ ( ( 𝑥 = 𝐹 ∧ 𝑧 = ∅ ) → ( ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
61 |
60
|
opelopabga |
⊢ ( ( 𝐹 ∈ V ∧ ∅ ∈ V ) → ( 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
62 |
42 61
|
mpan2 |
⊢ ( 𝐹 ∈ V → ( 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
63 |
59
|
elabg |
⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ↔ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝐹 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝐹 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
64 |
52 62 63
|
3bitr4d |
⊢ ( 𝐹 ∈ V → ( 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) |
65 |
64
|
adantl |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) |
66 |
65
|
orbi2d |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) ) |
67 |
|
eqid |
⊢ ( ∅ Sat ∅ ) = ( ∅ Sat ∅ ) |
68 |
67
|
satf0suc |
⊢ ( 𝑦 ∈ ω → ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) = ( ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∪ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) |
69 |
68
|
eleq2d |
⊢ ( 𝑦 ∈ ω → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∪ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) ) |
70 |
|
elun |
⊢ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∪ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) |
71 |
69 70
|
bitrdi |
⊢ ( 𝑦 ∈ ω → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ↔ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) ) |
72 |
71
|
ad2antrr |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ↔ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 〈 𝐹 , ∅ 〉 ∈ { 〈 𝑥 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) ) |
73 |
|
fmlasuc0 |
⊢ ( 𝑦 ∈ ω → ( Fmla ‘ suc 𝑦 ) = ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) |
74 |
73
|
eleq2d |
⊢ ( 𝑦 ∈ ω → ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ 𝐹 ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) ) |
75 |
74
|
ad2antrr |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ 𝐹 ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) ) |
76 |
|
elun |
⊢ ( 𝐹 ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ↔ ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ∨ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) |
77 |
76
|
a1i |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 𝐹 ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ↔ ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ∨ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) ) |
78 |
|
simpr |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) → ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) |
79 |
78
|
imp |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) |
80 |
79
|
orbi1d |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ∨ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ↔ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) ) |
81 |
75 77 80
|
3bitrd |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ ( 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ∨ 𝐹 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) } ) ) ) |
82 |
66 72 81
|
3bitr4rd |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) ) ∧ 𝐹 ∈ V ) → ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ) ) |
83 |
82
|
exp31 |
⊢ ( 𝑦 ∈ ω → ( ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑦 ) ) ) → ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ suc 𝑦 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ suc 𝑦 ) ) ) ) ) |
84 |
6 12 18 24 48 83
|
finds |
⊢ ( 𝑁 ∈ ω → ( 𝐹 ∈ V → ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) ) |
85 |
84
|
com12 |
⊢ ( 𝐹 ∈ V → ( 𝑁 ∈ ω → ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) ) |
86 |
|
prcnel |
⊢ ( ¬ 𝐹 ∈ V → ¬ 𝐹 ∈ ( Fmla ‘ 𝑁 ) ) |
87 |
86
|
adantr |
⊢ ( ( ¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω ) → ¬ 𝐹 ∈ ( Fmla ‘ 𝑁 ) ) |
88 |
|
opprc1 |
⊢ ( ¬ 𝐹 ∈ V → 〈 𝐹 , ∅ 〉 = ∅ ) |
89 |
88
|
adantr |
⊢ ( ( ¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω ) → 〈 𝐹 , ∅ 〉 = ∅ ) |
90 |
|
satf0n0 |
⊢ ( 𝑁 ∈ ω → ∅ ∉ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) |
91 |
|
df-nel |
⊢ ( ∅ ∉ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ↔ ¬ ∅ ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) |
92 |
90 91
|
sylib |
⊢ ( 𝑁 ∈ ω → ¬ ∅ ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) |
93 |
92
|
adantl |
⊢ ( ( ¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω ) → ¬ ∅ ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) |
94 |
89 93
|
eqneltrd |
⊢ ( ( ¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω ) → ¬ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) |
95 |
87 94
|
2falsed |
⊢ ( ( ¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω ) → ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) |
96 |
95
|
ex |
⊢ ( ¬ 𝐹 ∈ V → ( 𝑁 ∈ ω → ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) ) |
97 |
85 96
|
pm2.61i |
⊢ ( 𝑁 ∈ ω → ( 𝐹 ∈ ( Fmla ‘ 𝑁 ) ↔ 〈 𝐹 , ∅ 〉 ∈ ( ( ∅ Sat ∅ ) ‘ 𝑁 ) ) ) |